Exercise 1, § 1

Let $f(z) = z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n$ be a polynomial of degree $n$ with complex coefficients and put $f(z) = \prod_{i=1}^n (z - z_i)$. Let $r_0 = \sup_i |z_i|$.

a) Show that if the real number $r > 0$ is such that

then $r_0 \leq r$, and deduce that

b) Let $(\lambda_i)_{1 \leq i \leq n}$ be a finite sequence of $n$ real numbers $> 0$ such that $\sum_{i=1}^n \lambda_i^{-1} = 1$. Show that $r_0 \leq \sup_k (\lambda_k |a_k|)^{1/k}$ [use a)].

c) Deduce from a) that, if the coefficients $a_i$ are all non-zero, we have

d) Deduce from a) that

[consider the polynomial $(z - 1)f(z)$]. Hence show that, if the $a_i$ are all real and $> 0$, we have